Liouville theorems for an integral equation of Choquard type

Phuong Le

doi:10.3934/cpaa.2020036

Article Contents

2020, Volume 19, Issue 2: 771-783. Doi: 10.3934/cpaa.2020036

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Liouville theorems for an integral equation of Choquard type

Phuong Le^1,2,

1.
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2.
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

Received: October 31, 2018

Revised: June 30, 2019

Published: October 2019

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Abstract

We establish sharp Liouville theorems for the integral equation

$ u(x) = \int_{\mathbb{R}^n} \frac{u^{p-1}(y)}{|x-y|^{n-\alpha}} \int_{\mathbb{R}^n} \frac{u^p(z)}{|y-z|^{n-\beta}} dz dy, \quad x\in\mathbb{R}^n, $

where $ 0<\alpha, \beta<n $ and $ p>1 $. Our results hold true for positive solutions under appropriate assumptions on $ p $ and integrability of the solutions. As a consequence, we derive a Liouville theorem for positive $ H^{\frac{\alpha}{2}}(\mathbb{R}^n) $ solutions of the higher fractional order Choquard type equation

$ (-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * u^p\right) u^{p-1} \quad\text{ in } \mathbb{R}^n. $

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Mathematics Subject Classification: Primary: 35R11, 35J91; Secondary: 45G10, 35B53.

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